Es wird eine alternative Normalform für Elliptische Kurven gegeben und deren Modulraum studiert. Weiterhin wird die Konfiguration der Wendepunkte eingehend studiert. Für weitere Details verweisen wir auf die Einleitung.
Table of Contents
1 Hesse pencil and Hesse configuration
1.1 A geometrical approach
1.2 The Hesse configuration
1.3 The Hesse group
1.4 An algebraic approach
1.5 The Weierstrass normal form and the j-invariant
2 The Cayleyan curve
2.1 Polar hypersurfaces
2.2 Polar quadrics and the Hessian
2.3 The dual Hesse pencil and the Cayleyan of a plane cubic
Objectives & Core Topics
This thesis examines the configuration of inflection points on elliptic curves within the projective plane, focusing on the geometry of the Hesse pencil and its dual counterpart, the Cayleyan curve.
- The Hesse pencil as a linear system of cubic curves
- Properties of the Hessian group and its relation to F3
- Theoretical foundations of polar hypersurfaces and polar quadrics
- Characterization of the Cayleyan curve of a smooth cubic curve
Excerpt from the Thesis
1.1 A geometrical approach
Definition 1.1.1. The Hesse pencil is the one-dimensional linear system of plane cubic curves in P2 given by Et0,t1 : t0(Z0^3 + Z1^3 + Z2^3) + t1Z0Z1Z2 = 0, with [t0 : t1] ∈ P1. We denote by P the set of members of the Hesse pencil, namely P := {E1,λ : λ ∈ k} ∪ {E0,1}. In the following we use an affine parameter λ := t1/t0 and denote E1,λ by Eλ. The curve E0,1 is denoted by E∞.
Definition 1.1.2. Let E be a plane cubic curve defined by F(Z0, Z1, Z2) = 0. Then He(E), the Hessian curve of E (or just the Hessian), is the plane cubic curve defined to be V (det((∂i∂jF)i,j=1,2)), namely the zero locus of the determinant of the Hesse matrix of F.
A very important role in this first part of the thesis plays the following Lemma 1.1.3. Let E be a smooth cubic. Then the set of inflection points of E coincides with the intersection of E and He(E).
Proof. We refer to Proposition 2.5 in [Kuw11]. The proof uses the theory of polar hypersurfaces, which is discussed in Chapter 2.1 of this thesis. Note that we do not use this lemma to prove anything in the second chapter, hence the logic is not disturbed.
Summary of Chapters
1 Hesse pencil and Hesse configuration: This chapter introduces the Hesse pencil and the geometry of inflection points, providing both a geometric and an algebraic perspective on the configuration.
1.1 A geometrical approach: This section defines the Hesse pencil and the Hessian curve, establishing the fundamental relationship between smooth cubics and their inflection points.
1.2 The Hesse configuration: This section details the specific (94, 123) configuration formed by inflection points and the lines connecting them.
1.3 The Hesse group: This section identifies the Hessian group G216 and explores its isomorphism to the special affine linear group of F3^2.
1.4 An algebraic approach: This section constructs the configuration through a group law defined on the elliptic curve using the 3-torsion points.
1.5 The Weierstrass normal form and the j-invariant: This section computes the Weierstrass form and j-invariant for members of the Hesse pencil.
2 The Cayleyan curve: This chapter shifts focus to the theory of polar hypersurfaces and the construction of the dual Cayleyan curve.
2.1 Polar hypersurfaces: This section establishes the theoretical framework for polar hypersurfaces required for dual constructions.
2.2 Polar quadrics and the Hessian: This section provides an alternative description of the Hessian using polar quadrics.
2.3 The dual Hesse pencil and the Cayleyan of a plane cubic: This section connects the Steinerian automorphism to the dual pencil and defines the Cayleyan curve of a plane cubic.
Keywords
Hesse pencil, Hesse configuration, Elliptic curves, Inflection points, Hessian curve, Polar hypersurfaces, Polar quadrics, Cayleyan curve, j-invariant, Weierstrass normal form, 3-torsion points, Projective plane, Steinerian automorphism, Harmonic polars, Algebraic geometry
Frequently Asked Questions
What is the primary focus of this thesis?
The thesis focuses on the geometric and algebraic properties of the Hesse pencil and the construction of the Cayleyan curve of a plane cubic curve.
What are the main thematic areas covered?
The work covers the Hesse configuration, the Hessian group, polar hypersurface theory, and the construction of dual Hesse pencils.
What is the research goal of this paper?
The goal is to study the inflection points of elliptic curves and demonstrate how the Cayleyan curve relates to the dual Hesse pencil.
Which mathematical methods are primarily employed?
The author uses techniques from algebraic geometry, specifically linear systems of curves, the theory of polar hypersurfaces, and group actions on elliptic curves.
What topics are discussed in the main body?
The main body examines the geometric definition of the Hesse pencil, the classification of the Hessian group, and the Steinerian automorphism in the context of the Cayleyan curve.
Which keywords define this work?
Key terms include Hesse pencil, Hesse configuration, elliptic curves, inflection points, Hessian curve, polar quadrics, and the Cayleyan curve.
What is the significance of the (94, 123) configuration mentioned?
It refers to a specific geometric configuration of nine points and twelve lines where each point lies on four lines and each line contains three points, intrinsic to the Hesse pencil.
How is the Steinerian automorphism utilized?
It is used as a fixed-point-free involution that helps characterize the Cayleyan curve and relate it to the group law of the elliptic curve.
What role do harmonic polars play?
Harmonic polars are lines that intersect the cubic curve at specific points, playing a crucial role in defining the dual Hesse configuration.
Does the work rely on specific field characteristics?
Yes, while the first part considers arbitrary fields of characteristic not equal to three, the second part assumes a field of characteristic zero.
- Citation du texte
- Jakob Bongartz (Auteur), 2012, The Hesse pencil and its Cayleyan, Munich, GRIN Verlag, https://www.grin.com/document/335107
